IODS course project
Tina Lauronen
Chapter 1: About the project
The course description is available in mooc.helsinki.fi
Chapter 2: Regression and model validation
First look at data wrangling and analysis
Data wrangling
Reading the full data and naming it “learning2014”.
learning2014 <- read.table("http://www.helsinki.fi/~kvehkala/JYTmooc/JYTOPKYS3-data.txt", sep="\t", header = TRUE)Data consists of 183 rows (observations) and 60 columns (variables).
dim(learning2014)## [1] 183 60The structure of the data:
str(learning2014)## 'data.frame': 183 obs. of 60 variables:
## $ Aa : int 3 2 4 4 3 4 4 3 2 3 ...
## $ Ab : int 1 2 1 2 2 2 1 1 1 2 ...
## $ Ac : int 2 2 1 3 2 1 2 2 2 1 ...
## $ Ad : int 1 2 1 2 1 1 2 1 1 1 ...
## $ Ae : int 1 1 1 1 2 1 1 1 1 1 ...
## $ Af : int 1 1 1 1 1 1 1 1 1 2 ...
## $ ST01 : int 4 4 3 3 4 4 5 4 4 4 ...
## $ SU02 : int 2 2 1 3 2 3 2 2 1 2 ...
## $ D03 : int 4 4 4 4 5 5 4 4 5 4 ...
## $ ST04 : int 4 4 4 4 3 4 2 5 5 4 ...
## $ SU05 : int 2 4 2 3 4 3 2 4 2 4 ...
## $ D06 : int 4 2 3 4 4 5 3 3 4 4 ...
## $ D07 : int 4 3 4 4 4 5 4 4 5 4 ...
## $ SU08 : int 3 4 1 2 3 4 4 2 4 2 ...
## $ ST09 : int 3 4 3 3 4 4 2 4 4 4 ...
## $ SU10 : int 2 1 1 1 2 1 1 2 1 2 ...
## $ D11 : int 3 4 4 3 4 5 5 3 4 4 ...
## $ ST12 : int 3 1 4 3 2 3 2 4 4 4 ...
## $ SU13 : int 3 3 2 2 3 1 1 2 1 2 ...
## $ D14 : int 4 2 4 4 4 5 5 4 4 4 ...
## $ D15 : int 3 3 2 3 3 4 2 2 3 4 ...
## $ SU16 : int 2 4 3 2 3 2 3 3 4 4 ...
## $ ST17 : int 3 4 3 3 4 3 4 3 4 4 ...
## $ SU18 : int 2 2 1 1 1 2 1 2 1 2 ...
## $ D19 : int 4 3 4 3 4 4 4 4 5 4 ...
## $ ST20 : int 2 1 3 3 3 3 1 4 4 2 ...
## $ SU21 : int 3 2 2 3 2 4 1 3 2 4 ...
## $ D22 : int 3 2 4 3 3 5 4 2 4 4 ...
## $ D23 : int 2 3 3 3 3 4 3 2 4 4 ...
## $ SU24 : int 2 4 3 2 4 2 2 4 2 4 ...
## $ ST25 : int 4 2 4 3 4 4 1 4 4 4 ...
## $ SU26 : int 4 4 4 2 3 2 1 4 4 4 ...
## $ D27 : int 4 2 3 3 3 5 4 4 5 4 ...
## $ ST28 : int 4 2 5 3 5 4 1 4 5 2 ...
## $ SU29 : int 3 3 2 3 3 2 1 2 1 2 ...
## $ D30 : int 4 3 4 4 3 5 4 3 4 4 ...
## $ D31 : int 4 4 3 4 4 5 4 4 5 4 ...
## $ SU32 : int 3 5 5 3 4 3 4 4 3 4 ...
## $ Ca : int 2 4 3 3 2 3 4 2 3 2 ...
## $ Cb : int 4 4 5 4 4 5 5 4 5 4 ...
## $ Cc : int 3 4 4 4 4 4 4 4 4 4 ...
## $ Cd : int 4 5 4 4 3 4 4 5 5 5 ...
## $ Ce : int 3 5 3 3 3 3 4 3 3 4 ...
## $ Cf : int 2 3 4 4 3 4 5 3 3 4 ...
## $ Cg : int 3 2 4 4 4 5 5 3 5 4 ...
## $ Ch : int 4 4 2 3 4 4 3 3 5 4 ...
## $ Da : int 3 4 1 2 3 3 2 2 4 1 ...
## $ Db : int 4 3 4 4 4 5 4 4 2 4 ...
## $ Dc : int 4 3 4 5 4 4 4 4 4 4 ...
## $ Dd : int 5 4 1 2 4 4 5 3 5 2 ...
## $ De : int 4 3 4 5 4 4 5 4 4 2 ...
## $ Df : int 2 2 1 1 2 3 1 1 4 1 ...
## $ Dg : int 4 3 3 5 5 4 4 4 5 1 ...
## $ Dh : int 3 3 1 4 5 3 4 1 4 1 ...
## $ Di : int 4 2 1 2 3 3 2 1 4 1 ...
## $ Dj : int 4 4 5 5 3 5 4 5 2 4 ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ Attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...Acces the dplyr library:
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionCombinig the questions related to deep, surface and strategic learning.
deep_questions <- c("D03", "D11", "D19", "D27", "D07", "D14", "D22", "D30","D07","D14","D22","D30")
surface_questions <-c("SU02","SU10","SU18","SU26", "SU05","SU13","SU21","SU29","SU08","SU16","SU24","SU32")
strategic_questions <- c("ST01","ST09","ST17","ST25","ST04","ST12","ST20","ST28")Selecting the columns for “deep” and taking average to scale the variable.
deep_columns <- select(learning2014, one_of(deep_questions))
learning2014$deep <- rowMeans(deep_columns)Selecting the columns for “surf” and taking average to scale the variable.
surface_columns <- select(learning2014, one_of(surface_questions))
learning2014$surf <- rowMeans(surface_columns)Selecting the columns for “stra” and taking average to scale the variable.
strategic_columns <- select(learning2014, one_of(strategic_questions))
learning2014$stra <- rowMeans(strategic_columns)Selecting the variables gender, age, attitude towards statistics, exam points and the three new summary variables to dataset.
keep_columns <- c("gender", "Age", "Attitude", "deep", "stra", "surf", "Points")
new_learning2014 <- select(learning2014, one_of(keep_columns))Removing the observations with 0 exam points:
new_learning2014 <- filter(new_learning2014, Points > 0)Testing: The dimension and structure seem to be correct.
dim(new_learning2014)## [1] 166 7str(new_learning2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ Attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.75 2.88 3.88 3.5 3.75 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...Saving the dataset:
write.table(new_learning2014, "learning2014.txt", sep = "\t")Reading the set again and after that testing it is working properly:
str(new_learning2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ Attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.75 2.88 3.88 3.5 3.75 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...dim(new_learning2014)## [1] 166 7head(new_learning2014)## gender Age Attitude deep stra surf Points
## 1 F 53 37 3.750 3.375 2.583333 25
## 2 M 55 31 2.875 2.750 3.166667 12
## 3 F 49 25 3.875 3.625 2.250000 24
## 4 M 53 35 3.500 3.125 2.250000 10
## 5 M 49 37 3.750 3.625 2.833333 22
## 6 F 38 38 4.875 3.625 2.416667 21…and everything seems to be ok and I’m able move forward to the analysis part of the exercises.
Analysis
- Describing the data
I’m using the dataset from the previous part of the exercise. The data is part of a larger “Approaches to Learning” -survey. Here is used only a few backgroud variables (age and gender), variable measuring global attitude toward statistics, exam points and sum variables measuring student’s deep, surface and strategic learning.
When observations with zero exam points are excluded, the total N of the data is 166.
- Descriptive statistics
First thing to do with the data is to see some descriptive statistics.
summary(new_learning2014)## gender Age Attitude deep stra
## F:110 Min. :17.00 Min. :14.00 Min. :1.625 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:26.00 1st Qu.:3.500 1st Qu.:2.625
## Median :22.00 Median :32.00 Median :3.875 Median :3.188
## Mean :25.51 Mean :31.43 Mean :3.796 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:37.00 3rd Qu.:4.250 3rd Qu.:3.625
## Max. :55.00 Max. :50.00 Max. :4.875 Max. :5.000
## surf Points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00There are 110 female and 56 male respondents in the data. Tha average age is 25.5 when the youngest respondent is 17 and the oldest 55 years old. The measured mean attitude towards statistics is 31.4, ranging from 14 to 50. Exam points range (when 0 points excluded) from 7 to 33 and the average is 22.7. The summary variables measuring deep, strategic and surface learning are scaled from 1 to 5. In deep learning the averige is 3.8 (minimum 1.6, maximum 4.9), strategic learning 3.1 (min 1.3, max 5.0) and surface learning 2.8 (min 1.6, max 4.3).
As a graphical overview I made a scatter plot of all the variables according to background variables.
plot(new_learning2014$Attitude, new_learning2014$Age, col=new_learning2014$gender, title("Figure 1: Attitude according to age and gender"))
Males and older respondents seem to have more attitude towards statistics.
plot(new_learning2014$Points, new_learning2014$Age, col=new_learning2014$gender, title("Figure 2: Exam points according to age and gender"))
There is a lot of variation in exam points, no clear trends visible. Especially in the lower points there are both genders and respondents of all ages represent.
plot(new_learning2014$deep, new_learning2014$Age, col=new_learning2014$gender, title("Figure 3: Deep learning according to age and gender"))
In deep learning there is a clear impact of age: older respondets tend to use learning methods that are connected with deep learning.
plot(new_learning2014$stra, new_learning2014$Age, col=new_learning2014$gender, title("Figure 4: Strategic learning according to age and gender"))
In strategic learning the impact of age is not at visible as above, but some connection is visible.
plot(new_learning2014$surf, new_learning2014$Age, col=new_learning2014$gender, title("Figure 5: Surface learning according to age and gender"))
Surface learning is less usual than the other two, but surprisingly there are no clear differences according to age nor gender.
- Regression model
Creating a regression model with three explanatory variables (attitude, deep learning and strategic learning). The dependent variable is exam points. Checking the summary of the model.
rg_model <- lm(Points ~ Attitude + deep + stra, data = new_learning2014)
summary(rg_model)##
## Call:
## lm(formula = Points ~ Attitude + deep + stra, data = new_learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.525 -3.342 0.572 3.902 11.646
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.05207 3.43928 3.504 0.000592 ***
## Attitude 0.35116 0.05654 6.211 4.27e-09 ***
## deep -0.90243 0.72430 -1.246 0.214584
## stra 0.97852 0.53610 1.825 0.069805 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.28 on 162 degrees of freedom
## Multiple R-squared: 0.2124, Adjusted R-squared: 0.1978
## F-statistic: 14.56 on 3 and 162 DF, p-value: 1.925e-08The summary table tells us that when attitude rises one point the exam point increses by 0.35 points. The similar connecton with deep learning is -0.90 and with strategic learning 0.98.
The only statistically significant variable is the attitude. I tried surface learning and both age and gender as well, but they were not statistically significant in the analysis. In this case I remove all the other variables but the attitude.
- Summary of the fitted model
The final regression model is simple with only one independet, explanatory variable.
rg_model2 <- lm(Points ~ Attitude, data = new_learning2014)
summary(rg_model2)##
## Call:
## lm(formula = Points ~ Attitude, data = new_learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.63715 1.83035 6.358 1.95e-09 ***
## Attitude 0.35255 0.05674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09The effect of attitude stays the same as above: when the level of attitude grows one point, the exam points increse by 0.35 points. The effect is not very strong, but it is statistically significant, which means that there is very low change to make the wrong assumption about the relationship of these two variables.
- Diagnostic plots
Residuals vs. Fitted values:
In this diagnostic plot we examine if the errors are just errors and not depending on the explanatory variable. This means that there should not be any kind of pattern in the plot - and luckily there is not.
plot(rg_model2, which = 1)
Normal QQ-plot:
The plot shows that the residuals are close enough normally distributed. The residuals follow the line quite well.
plot(rg_model2, which = 2)
Residuals vs. Leverage:
In this plot we examine the impact of an individual observation on the model, basically by checking how far from the average are the (possible) observations that do not fit in the model. In this case there is nothing to worry.
plot(rg_model2, which = 5)
Chapter 3: Logistic regression
The data
The data set used in this exercise is joined dataset about student alcohol consumption. The data and some information about it is available here. The data set is joined and motified in the previous data wrangling exercise. The RScript is available here. We start by reading the dataset from the file.
alc <- read.csv("/Users/tlaurone/GitHub/tinalauronen/IODS-project/data/alc.csv")In the joined data set there are 35 variables and N = 382. The variable names are the following:
colnames(alc)## [1] "X" "school" "sex" "age" "address"
## [6] "famsize" "Pstatus" "Medu" "Fedu" "Mjob"
## [11] "Fjob" "reason" "nursery" "internet" "guardian"
## [16] "traveltime" "studytime" "failures" "schoolsup" "famsup"
## [21] "paid" "activities" "higher" "romantic" "famrel"
## [26] "freetime" "goout" "Dalc" "Walc" "health"
## [31] "absences" "G1" "G2" "G3" "alc_use"
## [36] "high_use"We also need some libraries to conduct the analyses.
library(tidyr); library(dplyr); library(ggplot2); library(gmodels)The variables and hyphothesis
The dependent variable is high alcohol consumption. It is a dichotomy variable (true/false) counted from the variables measuring weekly and daily alcohol consumption (scale from 1 - very low to 5 - very high). If the average of those alcohol consumption variables is more than 2, the respondent is classified in the high consumption category. From the 382 respondents 114 are in the high usage category and 268 are not.
The independent (explanatory) variables chosen to this analysis are sex, mother’s education, going out with friends and romantic relationship.
Distributions and preliminary analysis
There are 198 females and 184 males in the data. The assumption is that males consume more alcohol. Mother’s education is measured on a five step scale (0 = no education, 1 = primary education, 2 = 5th or 9th grade, 3 = secondary education, 4 = higher education). The assumption is that the more educated the mother is, the less the student uses alcohol.
Going out with friends is measured by a scale from 1 (very low) to 5 (very heigh). The assumption is that the more student goes out with friends, the more s/he uses alcohol. Romantic relationship is measured as a simple dichotomy, yes or no. The assumption is that romantic relationship decreases the alcohol consumption.
The connection between sex and high alcohol consumption is easiest to explore by cross tabulation. Below we can see, that 39.1 % of male respondents are considered high users, compared to only 21.2 % of females. Of all high users 63.2 % are male and 36.8 % female. The results are statistically significant (p<0.001).
CrossTable(alc$high_use, alc$sex, prop.r = TRUE, prop.c = TRUE, prop.t = FALSE, prop.chisq = FALSE, chisq = TRUE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## |-------------------------|
##
##
## Total Observations in Table: 382
##
##
## | alc$sex
## alc$high_use | F | M | Row Total |
## -------------|-----------|-----------|-----------|
## FALSE | 156 | 112 | 268 |
## | 0.582 | 0.418 | 0.702 |
## | 0.788 | 0.609 | |
## -------------|-----------|-----------|-----------|
## TRUE | 42 | 72 | 114 |
## | 0.368 | 0.632 | 0.298 |
## | 0.212 | 0.391 | |
## -------------|-----------|-----------|-----------|
## Column Total | 198 | 184 | 382 |
## | 0.518 | 0.482 | |
## -------------|-----------|-----------|-----------|
##
##
## Statistics for All Table Factors
##
##
## Pearson's Chi-squared test
## ------------------------------------------------------------
## Chi^2 = 14.62517 d.f. = 1 p = 0.000131151
##
## Pearson's Chi-squared test with Yates' continuity correction
## ------------------------------------------------------------
## Chi^2 = 13.78187 d.f. = 1 p = 0.0002053081
##
## To present the distribution of the other independent variables and their connections to the dependent variable, we use simple bar plots.
In Figure 1 there is the distribution of variable mother’s education presented. By adding the high alcohol use variable in the figure, it is possible to see that there is some connection between the variables: compared to primary education it is less likely that children of bit more educated mothers are less likely high users of alcohol. Secondary education increses the share of high drinkers, but again the higher education seems to have connection to less usage.
qplot(Medu, data = alc, color = high_use, main = "Figure 1: Mother's education")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Figure 2 presents the frequencies of the variable going out with friends and its connections to high alcohol usage. As assumed, the more students go out with friends the more likely they use more alcohol.
qplot(goout, data = alc, color = high_use, main = "Figure 2: Going out with friends")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
In Figure 3 there is no clear, visible diffenrence in high alcohol consumption an romantic relationships because the amount of students not in a relationship is so much higher.
qplot(romantic, data = alc, color = high_use, main = "Figure 3: Having romantic relationship")
Simple cross tabulations or bar plot are not sufficient methods to approach a question as complicated as high alcohol consumption. It is necessary to move forward to more sophisticated methods.
Logistic regression analysis
Using the variables presented above we conduct a logistic regression analysis to explore high alcohol usage. First we make the model and print the summary of it, then we compute and print odds ratios (OR) and confidence intervals (CI).
r_model <- glm(high_use ~ sex + Medu + goout + romantic, data = alc, family = "binomial")
summary.glm(r_model)##
## Call:
## glm(formula = high_use ~ sex + Medu + goout + romantic, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.6002 -0.8624 -0.6150 0.8460 2.5276
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -3.63037 0.54119 -6.708 1.97e-11 ***
## sexM 0.87723 0.24830 3.533 0.000411 ***
## Medu -0.05562 0.11326 -0.491 0.623384
## goout 0.76382 0.11819 6.463 1.03e-10 ***
## romanticyes -0.11913 0.26591 -0.448 0.654154
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 402.30 on 377 degrees of freedom
## AIC: 412.3
##
## Number of Fisher Scoring iterations: 4OR <- coef(r_model) %>% exp
CI <- confint(r_model) %>% exp## Waiting for profiling to be done...cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.02650637 0.008790626 0.07374126
## sexM 2.40422513 1.484855219 3.93774659
## Medu 0.94590023 0.757602427 1.18226671
## goout 2.14645372 1.712829640 2.72499480
## romanticyes 0.88769573 0.523308804 1.48804026From the coefficient table from the summary we can see, that only sex and going out with friends are statistically significant variables explaining the high alcohol usage. From the OR table we are able to interpret that if a student is a male (versus being a female) he is 2.4 times more likely a high user. Going out with friends adds the probability to be a high user as well, but due to the measurement scale of the variable the interpretation is not as straightforward.
If not considering the statistical significance levels it is possible to simplify the interpretation of the OR’s: If the OR is less than 1, it is less likely to be a high user. This would mean in our case that more educated mother and being in a romantic relationship would decrese the probability to use more alcohol. Insted, being a male and going out with friends a lot increses that probability - and the effect is stronger as well. These results are in line with the preassumptions - but there is no reason to claim that in case of mother’s education and romantic relationship.
Predictions
First we create a new regression model with the statistically significant variables, sex and going out with friends from the previous model. Then we make a new variable: first one that contains the prediction, then one that predicts high alcohol usage.
r_m <- glm(high_use ~ sex + goout, data = alc, family = "binomial")
probabilities <- predict(r_m, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = (probability > 0.5))Then we make a table of the original variable, high alcohol usage, and the variable predicting it.
table(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 251 17
## TRUE 63 51…and here is the same comparison in a graphical form:
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
Here we compute the proportion of inaccurately classified individuals. First we create a function to claculate the training error, the we do the actual math.
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2094241When testing the function by giving the parameter prob values 0 or 1 (meaning that none of our respondents are high users or that all of them are) the result of the actual variable probability is better (the value of loss_func is smaller). By plain guessing it would have gone (more) wrong.
K-fold cross-validation
Finally we test our model with K-fold cross-validation.
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = r_m, K = 10)
cv$delta[1]## [1] 0.2303665The cross-validation exercise in DataCamp resulted error value of 0.26. Here the error value is approximately 0.23 which means that our model is better.
Chapter 4: Clustering and classification
Data
In this exercise we use dataset “Boston” from the MASS package for R. First we take a look at the structure and dimensions of the data. The data has 14 variables and 506 observations. Detailed information about variables in found here. Apparently the observations are towns of Boston.
##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select## Loading tidyverse: tibble
## Loading tidyverse: readr
## Loading tidyverse: purrr## Conflicts with tidy packages ----------------------------------------------## filter(): dplyr, stats
## lag(): dplyr, stats
## select(): dplyr, MASS## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14Graphical overview and summaries
The graphical overview of all the variables does not tell much.
Instead we take a look at the summaries of the variables:
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Then we check the correlations between the variables. There are strong negative correlations between weighted mean of distances to five Boston employment centres and proportion of non-retail business acres per town, nitrogen oxides concentration (parts per 10 million) and proportion of owner-occupied units built prior to 1940. Strong negative correlation is found as well between lower status of the population (percent) and median value of owner-occupied homes in $1000s. Strong negative correlations are found especially between index of accessibility to radial highways and full-value property-tax rate per $10,000.
Modifying the dataset
First we standardize the dataset and print out the summaries of the scaled data. From the summary we can see, that the means of the scaled variables are zero. This means that the variables are centered.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865From the scaled variable of the per capita crime rate by town we create a categorical variable of the crime rate. We use quantiles as a base of categorization. As a result we get a variable that divides the towns of Boston to low, medium low, medium high and high crime rates.
## crime
## low med_low med_high high
## 127 126 126 127Finally we divide the dataset to train (80 per cent) and test (20 per cent) datasets.
Linear Discriminant Analysis (LDA)
Next we conduct LDA on the train set using the categorical crime rate variable as target variable. As a result we draw a LDA (bi)plot.
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2500000 0.2376238 0.2524752 0.2599010
##
## Group means:
## zn indus chas nox rm
## low 0.8890734 -0.9019751 -0.11640431 -0.8485819 0.49436000
## med_low -0.1089388 -0.3553339 0.05576262 -0.5928329 -0.12540763
## med_high -0.3939194 0.2094853 0.19085920 0.3787043 0.07552006
## high -0.4872402 1.0170492 -0.04735191 1.0600685 -0.40918594
## age dis rad tax ptratio
## low -0.8691996 0.8161715 -0.6998714 -0.7442585 -0.41072487
## med_low -0.3743086 0.3381099 -0.5523924 -0.4877614 -0.09878629
## med_high 0.4282763 -0.3891904 -0.4177712 -0.3039529 -0.20362070
## high 0.8215287 -0.8583643 1.6388211 1.5145512 0.78158339
## black lstat medv
## low 0.3789216 -0.78701086 0.57711511
## med_low 0.3525325 -0.15404466 0.02871215
## med_high 0.1123583 0.03276762 0.12403068
## high -0.8229299 0.88127269 -0.68173081
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.089817591 0.59533732 -0.95914789
## indus 0.012756117 -0.38341216 0.09901529
## chas -0.092534062 -0.04440463 0.16581539
## nox 0.418771864 -0.68463165 -1.31335142
## rm -0.099805509 -0.08434489 -0.19921534
## age 0.250766268 -0.39682760 -0.13046924
## dis -0.062207153 -0.23696215 0.11078385
## rad 3.154057574 0.88330560 -0.13127986
## tax -0.001347437 0.15442029 0.76048458
## ptratio 0.141993361 -0.03173391 -0.38682081
## black -0.157804451 0.01308245 0.17129648
## lstat 0.204062083 -0.19696556 0.33832365
## medv 0.180249256 -0.31513496 -0.23749326
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9526 0.0349 0.0125## function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choises = c(1,2)){
## heads <- coef(x)
## arrows(x0 = 0, y0 = 0,
## x1 = myscale * heads[,choises[1]],
## y1 = myscale * heads[,choises[2]], col = color, length = arrow_heads)
## text(myscale * heads[,choises], labels = row.names(heads),
## cex = tex, col = color, pos = 3)
## }
Predicting the classes with LDA model
The categorical crime rate variable is removed from the test set. Now we predict the classes in the test data with the LDA model. From the crosstabulation on the correct and predicted obseravtions we are able to see that the prediction is mainly usable. The high rates all all prideicted right, and almost all the medium high observations as well. The medium low predictions are the most problematic.
## predicted
## correct low med_low med_high high
## low 16 10 0 0
## med_low 5 16 9 0
## med_high 1 5 17 1
## high 0 0 0 22K-means
Finally we reload the Boston dataset and standardize it. Then we calculate the distances between the observations. Summary of the Euclidean distances is presented below.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4620 4.8240 4.9110 6.1860 14.4000Then we run k-means algorithm, use WCSS to find out the optimal number of clusters, and run the algorithm again. First we try k-means with four clusters. From the WCSS plot we find out that the optimal number of clusters is two.
From the later figure it would be possible to ivestigate the observations divided to two clusters according to every variable within the analysis. Personally I do not find this kind of giant figure matrixes very informative, bacause their readability is very low. Maybe this would be helpful when making decisions of further analysis. If the colours in the figure are very mixed, the variables are not very significant considering the clusters. If the colours seems to be well separated, the variables in the figure are affecting in the formulation of the clusters.
Chapter 5: Dimensionality reduction techniques
The data
The data used in this analysis is a combination of two data set collected by UNDP. There are more information of the data in this web page. The observations are countries (total 155) and variables used here are Life expectancy at birth, Expected years of schooling, Gross national income (GNI) per capita, Maternal mortality ratio, Adolescent birth rate, Share of seats in parliament (female), Population with at least some secondary education (female/male ratio) and Labour force participation rate (female/male ratio).
Some libraries are needed to conduct the analysis.
First we read the data and explore it. As supposed, there are 155 observations (countries) and 8 variables.
## [1] 155 8## 'data.frame': 155 obs. of 8 variables:
## $ Educ_sec_FM_ratio: num 1.007 0.997 0.983 0.989 0.969 ...
## $ Labour_FM_ratio : num 0.891 0.819 0.825 0.884 0.829 ...
## $ Educ_e : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ Life_e : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ MM_ratio : int 4 6 6 5 6 7 9 28 11 8 ...
## $ Ad_birth_rate : num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Rep_parl : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...Overview
Summary of the data and explanations for the variable names:
## Educ_sec_FM_ratio Labour_FM_ratio Educ_e Life_e
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI MM_ratio Ad_birth_rate Rep_parl
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50| Label | Variable |
|---|---|
| Educ_sec_FM_ratio | Population with at least some secondary education (female/male ratio) |
| Labour_FM_ratio | Labour force participation rate (female/male ratio) |
| Educ_e | Expected years of schooling |
| Life_e | Life expectancy at birth |
| GNI | Gross national income (GNI) per capita |
| MM_ratio | Maternal mortality ratio |
| Ad_birth_rate | Adolescent birth rate |
| Rep_parl | Share of seats in parliament (female) |
As expected, the observations (countries) differ each other radically. It is more interesting to investigate correlations between the variables. Strong positive correlations are found between Expected years of schooling and Life expectancy at birth, and between Maternal mortality ratio and Adolescent birth rate. An obvious strong negative correlation is found between Life expectancy at birth and Maternal mortality ratio. All in all the pattern looks quite clear: the more educated women (in addition to mens’ education), the higher education expectancy, life expectancy and GNI and lower maternal mortality and adolescent birth rate.
Principal component analysis (PCA)
Then we conduct a principal component analysis.
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000
## PC8
## Standard deviation 0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion 1.0000## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
The results are almost unreadable, hopefully because of use of unstandardized data. We must scale the data and conduct the analysis again.
Principal component analysis (PCA) with standardized data
First we scale the data and check the summary of it.
## Educ_sec_FM_ratio Labour_FM_ratio Educ_e Life_e
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## GNI MM_ratio Ad_birth_rate Rep_parl
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850The means are 0.0 so we can move on to PCA analysis.
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
## PC7 PC8
## Standard deviation 0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion 0.98702 1.00000
Now the results make sense and are much easier to interpret. From the summary we can see, that the first principal component (PC1) covers 53.6 per cent of the variation. PC2 covers 16.2 per cent. With unstandardized data some of the variables dominated the analysis with their scale differing the others, but now be are able to see how the countries are located according to principal components.
From the arrows of the biplot we can interpret the content of the principal components: PC2 consists of variables considering Labour force participation rate (female/male ratio) and Share of seats in parliament (female). It can be interpret as some kind of gender equality component. The more important PC1 consists on the other hand Maternal mortality ratio and Adolescent birth rate, and on the other hand Expected years of schooling, Life expectancy at birth, Gross national income (GNI) per capita and Population with at least some secondary education (female/male ratio). The variables are more related to health issues, especially to women’s health, and the variables correlated to them.
According to this analysis the old thesis about girls’ education as the best investment for overall well-being gets support.
MCA analysis with tea data
For MCA analysis we load the “Tea” data set from R package “FactoMineR”. First we explore the data set. There are many variables we are not interested in, so we reduce the amount of them.
## [1] 300 36## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...## breakfast tea.time evening lunch
## breakfast :144 Not.tea time:131 evening :103 lunch : 44
## Not.breakfast:156 tea time :169 Not.evening:197 Not.lunch:256
##
##
##
##
##
## dinner always home work
## dinner : 21 always :103 home :291 Not.work:213
## Not.dinner:279 Not.always:197 Not.home: 9 work : 87
##
##
##
##
##
## tearoom friends resto pub
## Not.tearoom:242 friends :196 Not.resto:221 Not.pub:237
## tearoom : 58 Not.friends:104 resto : 79 pub : 63
##
##
##
##
##
## Tea How sugar how
## black : 74 alone:195 No.sugar:155 tea bag :170
## Earl Grey:193 lemon: 33 sugar :145 tea bag+unpackaged: 94
## green : 33 milk : 63 unpackaged : 36
## other: 9
##
##
##
## where price age sex
## chain store :192 p_branded : 95 Min. :15.00 F:178
## chain store+tea shop: 78 p_cheap : 7 1st Qu.:23.00 M:122
## tea shop : 30 p_private label: 21 Median :32.00
## p_unknown : 12 Mean :37.05
## p_upscale : 53 3rd Qu.:48.00
## p_variable :112 Max. :90.00
##
## SPC Sport age_Q frequency
## employee :59 Not.sportsman:121 15-24:92 1/day : 95
## middle :40 sportsman :179 25-34:69 1 to 2/week: 44
## non-worker :64 35-44:40 +2/day :127
## other worker:20 45-59:61 3 to 6/week: 34
## senior :35 +60 :38
## student :70
## workman :12
## escape.exoticism spirituality healthy
## escape-exoticism :142 Not.spirituality:206 healthy :210
## Not.escape-exoticism:158 spirituality : 94 Not.healthy: 90
##
##
##
##
##
## diuretic friendliness iron.absorption
## diuretic :174 friendliness :242 iron absorption : 31
## Not.diuretic:126 Not.friendliness: 58 Not.iron absorption:269
##
##
##
##
##
## feminine sophisticated slimming
## feminine :129 Not.sophisticated: 85 No.slimming:255
## Not.feminine:171 sophisticated :215 slimming : 45
##
##
##
##
##
## exciting relaxing effect.on.health
## exciting :116 No.relaxing:113 effect on health : 66
## No.exciting:184 relaxing :187 No.effect on health:234
##
##
##
##
## ## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
## ## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...## Warning: attributes are not identical across measure variables; they will
## be dropped
Then we conduct a multiple correspondence analysis MCA, print the summary of it and visualize it.
##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |
The MDA biplot can be interpretet as a taste map of tea consumption. The first dimension tells us about experimentality or safety of the respondents’s tea drinking habits: on the other side of the scale there are people who drink Earl Grey, buy their tea from the chain store in tea bags (and this is very common, close to zero) and on the other side there are specialized tea shops and unpackaged green or black tea.
The second dimension is a bit harder to interpret, but on the upper side of the figure there are some traces of omnivorousness. Everything goes: the chain stores and the tea shops, unpacked and teapacks, and even “other” things than milk or lemon to add to tea. On the lower part of the figure the choises can be interpret more traditional.
As an overall interpretation it is easiest to say that the closer to the zero point of the two dimensions a variable is, the more common or “mainstream” it is.